Transfer matrix method for enumeration and generation of compact self-avoiding walks. I. Square lattices

1998 ◽  
Vol 109 (12) ◽  
pp. 5134-5146 ◽  
Author(s):  
A. Kloczkowski ◽  
R. L. Jernigan
Keyword(s):  
Transfer Matrix ◽  
Transfer Matrix Method ◽  
Matrix Method ◽  
Self Avoiding Walks

The Umbral Transfer-Matrix Method. IV. Counting Self-Avoiding Polygons and Walks

10.37236/1572 ◽  
2001 ◽  
Vol 8 (1) ◽  
Author(s):  
Doron Zeilberger
Keyword(s):  
Transfer Matrix ◽  
Cross Section ◽  
Polynomial Time ◽  
Transfer Matrix Method ◽  
Matrix Method ◽  
Vertical Cross Section ◽  
Trivial Case ◽  
Exponential Time ◽  
Self Avoiding Walks ◽  
Special Case

This is the fourth installment of the five-part saga on the Umbral Transfer-Matrix method, based on Gian-Carlo Rota's seminal notion of the umbra. In this article we describe the Maple packages USAP, USAW, and MAYLIS. USAP automatically constructs, for any specific $r$, an Umbral Scheme for enumerating, according to perimeter, the number of self-avoiding polygons with $\leq 2r$ horizontal edges per vertical cross-section. The much more complicated USAW does the analogous thing for self-avoiding walks. Such Umbral Schemes enable counting these classes of self-avoiding polygons and walks in polynomial time as opposed to the exponential time that is required by naive counting. Finally MAYLIS is targeted to the special case of enumerating classes of saps with at most two horizontal edges per vertical cross-section (equivalently column-convex polyominoes by perimeter), and related classes. In this computationally trivial case we can actually automatically solve the equations that were automatically generated by USAP. As an example, we give the first fully computer-generated proof of the celebrated Delest-Viennot result that the number of convex polyominoes with perimeter $2n+8$ equals $(2n+11)4^n-4(2n+1)!/n!^2$.

The transfer matrix method for lattice proteins—an application with cooperative interactions

Polymer ◽  
2004 ◽  
Vol 45 (2) ◽  
pp. 707-716 ◽  
Author(s):  
Andrzej Kloczkowski ◽  
Taner Z. Sen ◽  
Robert L. Jernigan
Keyword(s):  
Transfer Matrix ◽  
Transfer Matrix Method ◽  
Matrix Method ◽  
Cooperative Interactions ◽  
Lattice Proteins

A Modified Transfer Matrix Method for Asymmetric Rotor-Bearing Systems

1994 ◽  
Vol 116 (3) ◽  
pp. 309-317 ◽  
Author(s):  
Yuan Kang ◽  
An-Chen Lee ◽  
Yuan-Pin Shih
Keyword(s):  
Transfer Matrix ◽  
Transfer Matrix Method ◽  
Matrix Method ◽  
Harmonic Balance Method ◽  
Continuous System ◽  
Euler Beam ◽  
Rotating Shaft ◽  
Rotor Bearing ◽  
Asymmetric Rotor ◽  
Steady State Responses

A modified transfer matrix method (MTMM) is developed to analyze rotor-bearing systems with an asymmetric shaft and asymmetric disks. The rotating shaft is modeled by a Rayleigh-Euler beam considering the effects of the rotary inertia and gyroscopic moments. Specifically, a transfer matrix of the asymmetric shaft segments is derived in a continuous-system sense to give accurate solutions. The harmonic balance method is incorporated in the transfer matrix equations, so that steady-state responses of synchronous and superharmonic whirls can be determined. A numerical example is presented to demonstrate the effectiveness of this approach.

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